#8932070

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,5)(3,8,11,4)(6,9,15,13)(10,14,16,12)(18,22)(19,27)(20,30)(26,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,7,11)(2,4,5,8)(6,10,15,16)(9,12,13,14)(17,18,23,22)(19,28,27,24)(20,25,30,31)(21,32,29,26)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(1,4,7,8)(2,3,5,11)(6,14,15,12)(9,16,13,10)(17,19,23,27)(18,24,22,28)(20,29,30,21)(25,32,31,26)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,7,2)(3,8,11,4)(6,13,15,9)(10,14,16,12)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3)(2,8)(4,5)(6,10)(7,11)(9,14)(12,13)(15,16)(17,21,23,29)(18,26,22,32)(19,20,27,30)(24,25,28,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,6,7,15)(2,9,5,13)(3,10,11,16)(4,12,8,14)(17,22,23,18)(19,28,27,24)(20,31,30,25)(21,32,29,26)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839266);